1. Chapter 9 – Public Key Cryptography and RSA Every Egyptian received two names, which were known respectively as the true name and the good name, or the great name and the little name; and while the good or little name was made public, the true or great name appears to have been carefully concealed. —The Golden Bough, Sir James George Frazer

2. Private-Key Cryptography traditional private/secret/single key cryptography uses one key shared by both sender and receiver if this key is disclosed, communications are compromised also is symmetric, parties are equal hence does not protect sender from receiver forging a message & claiming is sent by sender

3. Public-Key Cryptography probably most significant advance in the 3000 year history of cryptography uses two keys – a public & a private key –Anyone knowing the public key can encrypt messages or verify signatures –But cannot decrypt messages or create signatures asymmetric since parties are not equal complements rather than replaces private key crypto

4. Public-Key Cryptography public-key/two-key/asymmetric cryptography involves the use of two keys: –a public-key, which may be known by anybody, and can be used to encrypt messages, and verify signatures –a private-key, known only to the recipient, used to decrypt messages, and sign (create) signatures is asymmetric because –those who encrypt messages or verify signatures cannot decrypt messages or create signatures

6. Why Public-Key Cryptography? developed to address two key issues: –key distribution – how to have secure communications in general without having to trust a KDC with your key No need for secure key delivery No one else needs to know your private key –digital signatures – how to verify a message comes intact from the claimed sender

7. Public-Key Characteristics Public-Key algorithms rely on two keys with the characteristics that it is: –computationally infeasible to find decryption key knowing only algorithm & encryption key –computationally easy to en/decrypt messages when the relevant (en/decrypt) key is known –Oneway-ness is desirable: exp/log, mul/fac –either of the two related keys can be used for encryption, with the other used for decryption (in some schemes)

8. Public-Key Cryptosystems:Secrecy

9. Public-Key Cryptosystems: Authentication

10. Public-Key Cryptosystems: Secrecy and Authentication

11. Public-Key Applications can classify uses into 3 categories: –encryption/decryption (provide secrecy) –digital signatures (provide authentication) –key exchange (of session keys) some algorithms are suitable for all uses, others are specific to one

12. Security of Public Key Schemes like private key schemes brute force exhaustive search attack is always theoretically possible but keys used are too large (>512bits) security relies on a large enough difference in difficulty between easy (en/decrypt) and hard (cryptanalyse) problems requires the use of very large numbers hence is slow compared to private key schemes

13. RSA by Rivest, Shamir & Adleman of MIT in 1977 best known & widely used public-key scheme based on exponentiation of integers in a finite (Galois) field –Defined over integers modulo a prime –exponentiation takes O((log n) 3 ) operations (easy) uses large integers (eg. 1024 bits) security due to cost of factoring large numbers –factorization takes O(e log n log log n ) operations (hard)

14. RSA Key Setup each user generates a public/private key pair by: 1. selecting two large primes at random - p, q (secret) 2. computing their system modulus N=p.q (public) –note ø(N)=(p-1)(q-1) (secret) 3. selecting at random the encryption key e (public) –where 1< e<ø(N), gcd(e,ø(N))=1 4. solve following equation to find decryption key d (secret) –e.d=1 mod ø(N) and 0≤d≤N –Use the extended Euclid’s algorithm to find the multiplicative inverse of e (mod ø(N)) publish their public encryption key: KU={e,N} keep secret private decryption key: KR={d,p,q}

15. Block size of RSA Each block is represented as an integer number Each block has a value M less than N The block size is <= log 2 (N) bits If the block size is k bits then 2 k <= N <= 2 K+1

16. RSA Use to encrypt a message M the sender: –obtains public key of recipient KU={e,N} –computes: C=M e mod N, where 0≤M<N to decrypt the ciphertext C the owner: – uses their private key KR={d,p,q} –computes: M=C d mod N note that the message M must be smaller than the modulus N (block if needed)

17. Why RSA Works because of Euler's Theorem: a ø(n) mod N = 1 –where gcd(a,N)=1 in RSA have: –N=p.q –ø(N)=(p-1)(q-1) –carefully chosen e & d to be inverses mod ø(N) –hence e.d=1+k.ø(N) for some k Two cases: –1. gcd(M, N) = 1 –2. gcd(M, N) > 1, see equation (8.6) in P.243

18. RSA Example 1.Select primes: p=17 & q=11 2.Compute n = pq =17×11=187 3.Compute ø(n)=(p–1)(q-1)=16×10=160 4.Select e : gcd(e,160)=1; choose e=7 5.Determine d : de=1 mod 160 and d < 160 Value is d=23 since 23×7=161= 10×160+1 6.Publish public key KU={7,187} 7.Keep secret private key KR={23,17,11}

19. RSA Example cont sample RSA encryption/decryption is: given message M = 88 ( 88<187 ) encryption: C = 88 7 mod 187 = 11 decryption: M = 11 23 mod 187 = 88

20. Exponentiation can use the Square and Multiply Algorithm a fast, efficient algorithm for exponentiation concept is based on repeatedly squaring base and multiplying in the ones that are needed to compute the result look at binary representation of exponent

21. Exponentiation

22. RSA Key Generation users of RSA must: –determine two primes at random - p, q –select either e or d and compute the other primes p,q must not be easily derived from modulus N=p.q –means must be sufficiently large –typically guess and use probabilistic test exponents e, d are inverses, so use Inverse algorithm to compute the other

23. RSA Security three approaches to attacking RSA: –brute force key search (infeasible given size of numbers) –mathematical attacks (based on difficulty of computing ø(N), by factoring modulus N) –timing attacks (on running of decryption)

24. Factoring Problem mathematical approach takes 3 forms: –factor N=p.q, hence find ø(N) and then d –determine ø(N) directly and find d –find d directly currently believe all equivalent to factoring –have seen slow improvements over the years as of Aug-99 best is 130 decimal digits (512) bit with GNFS –biggest improvement comes from improved algorithm cf “Quadratic Sieve” to “Generalized Number Field Sieve” –barring dramatic breakthrough 1024+ bit RSA secure ensure p, q of similar size and matching other constraints

25. Timing Attacks developed in mid-1990’s exploit timing variations in operations –infer bits of d based on time taken countermeasures –use constant exponentiation time –add random delays –blind values used in calculations C’ = (Mr) e, M’ = (C’) d, M=M’r -1

26. Summary have considered: –principles of public-key cryptography –RSA algorithm, implementation, security